Question

 The solution of the differential equation \[ (x-1)\frac{dx}{dy}+(y-2)=0, \] given that \(x=1\) when \(y=1\), represents a: 

• A. Parabola
• B. Circle
• C. Ellipse
• D. Hyperbola

Hint:

The general form \( (x-h)^2 + (y-k)^2 = r^2 \) represents a circle.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a separable differential equation: \( (x - 1) dx = -(y - 2) dy \).

Step 2: Detailed Explanation:
Integrate both sides:
\[ \int (x - 1) dx = -\int (y - 2) dy \]
\[ \frac{(x - 1)^2}{2} = -\frac{(y - 2)^2}{2} + C \]
\[ (x - 1)^2 + (y - 2)^2 = 2C \]
Given \( x = 1, y = 1 \):
\[ (1 - 1)^2 + (1 - 2)^2 = 2C \implies 0 + 1 = 2C \implies 2C = 1 \]
The equation becomes \( (x - 1)^2 + (y - 2)^2 = 1 \).

Step 3: Final Answer:
This is the equation of a circle with center \( (1, 2) \) and radius \( 1 \).